When an incoming wavefront from a distant star reaches the earth's atmosphere it has a plain wavefront and accurate imaging of the star is possible, as evidenced by the remarkably sharp pictures from the exo-terrestrial Space Telescope. However as it passes through the atmosphere the wavefront encounters turbulent layers of air resulting from thermal differences in the atmospheric layers and the wavefront is distorted, as shown schematically in FIG. 1. The farther the light travels through the air and the denser the air is, the greater the amplitude of the distortion typically becomes and the closer together the peaks in the wavefront distortion become. This distortion can be corrected by reflecting the wavefront from a mirror surface which perfectly matches the distorted wavefront laterally but has half the amplitude of the wavefront distortion. The phase of the light depends on the wavelength, so the shorter the wavelength the greater the phase error and the more critical its correction becomes. However in the visible region of the spectrum as much as 96% of the turbulence-induced atmospheric distortion can be removed using adaptive optics. Sharp pictures can be obtained from the ground as well as from space, as seen in FIG. 2. Adaptive optics has clearly opened a new chapter in the field of astronomy.
A statistical measure involving the phase error is a coefficient r0, often called the Fried (freed) coefficient after Professor Fried, the Professor at the Naval Postgraduate School in Monterey Calif. who first suggested it. As the Fried coefficient becomes smaller the distortion becomes greater. For zonal compensation involving actuators pushing against the flexible faceplate of mirror and affecting both displacement and local tip-tilt of the faceplate elements, the phase fitting error σF2 of the adaptive optic mirror to the wavefront, is given asσF2≈35(rθ/r0)5/3  (1)where r8 is the distance between actuators. If more actuators are used, for example let the actuator separation be decreased by a factor of 2, then the phase fitting error is decreased by a factor of about 3. Equation (1) implicitly assumes that the influence function of the faceplate, defined as the elementary deformation of the faceplate surface produced by one actuator, all other actuators acting only as springs, extends approximately to the next actuator. If the faceplate is not sufficiently flexible, so the influence function extends over many actuator separations, the mirror faceplate cannot distort to correct for peaks in the phase with spatial separations of the order of r0 and Eq. (1) does not represent the situation. As r0 becomes shorter the separation between actuators should not become significantly less than the faceplate influence function. If the faceplate is not sufficiently flexible, the fitting error can become unacceptably large.
The other important factor in adaptive optic operation is the time delay τ between the time the wavefront error is sensed and the time the actuator has moved to correct it. This time delay arises both from the sensing circuit τc, and the time constant of the actuator τa. The relationship between the mean squared phase error στ, and the delay time τ is again a 5/3 ds power dependence, and is given byστ2=(τ/τ0)5/3=6.88(τv/ro)5/3  (2)where the Greenwood time delay τo, the delay which results in a phase error of one radian, is given by τo=0.314 r0/v. Here v is the modulus of the average propagation velocity through a turbulence layer. Two delay times contribute to τ, the time constant of the feedback control circuit τc and the time constant τa, representing the response of the actuator itself in practice it is desirable that τc+τa<one millisec.